Particle Filters for Positioning in Wireless Networks Per-Johan Nordlund, Fredrik Gunnarsson, Fredrik Gustafsson Department of Electrical Engineering, Link¨opings universitet∗, SE-581 83 Link¨oping, Sweden Tel: +46 13 282226; fax: +46 13 282622 e-mail:
[email protected]
ABSTRACT Positioning of road vehicles in wireless radio networks is a highly non-linear multi-sensor problem. The radio measurements deliver snapshot information of position as circle, radial and hyperbolic lines with various reliability, and possibly also complex power attenuation maps. Further, fading and absence of line of sight give complicated disturbances on these measurements. Temporal and spatial prior knowledge may include maximal velocity and acceleration of the mobile unit, and for automotive applications also the constraint that most of the time is spent on roads, which are stored in a digital map. We outline a framework where all this information can be incorporated, and the true a posteriori distribution of position can be approximated with arbitrary accuracy to be traded off with real-time requirements. The algorithm is based on the particle filter, and we demonstrate it on a few simulation examples. 1
INTRODUCTION
The field of positioning in wireless networks is both driven by new market opportunities and laws for emergency call positioning. Various methods have been discussed in literature, and the aim here is to illuminate the potential of using particle filters for positioning. The position is determined using measurements, which are either network-assisted or mobile-assisted. The former requires less user equipment modifications, but is relatively costly, while the latter is cheaper for the operators but requires dedicated routines in the user equipment. For a more thorough overview, see [6, 16]. Proposed approaches utilize different types of measurements, which include: • Distance to base stations of known positions [7, 17, 18]. It is determined from time of arrival (TOA), time difference of arrival (TDOA) or enhanced observed time difference (E-OTD) measurements, see Figure 1. • Received Signal Strength of signals with known powers, which are closely related to the travelled distance [17, 14]. • Angle, which relates the angle of arrival (AOA) at the receiver to a fixed coordinate system. • Map Information, which is further described in [10]. ∗ The affiliations of the first two authors are also Saab Gripen and Ericsson Radio Systems, respectively. This project is supported by the competence center ISIS, which is acknowledged.
p1
p3
t1
t3
p t2 p2
Figure 1: Distance measurements in wireless networks. In uplink TOA, the base stations pi monitor the time of arrival ti of a characteristic burst from the mobile p to extract the distance |p − pi |. The mobile is more active in TDOA, where the time differences ∆t = ti − t1 are measured, an possible locations describe hyperbolas with the other base stations pi in foci. The effect of non-synchronized base stations is considered in E-OTD, where the network estimates the difference in transmission times between the base stations. When regular samples of the measurements are assumed available, a popular solution to this sensor fusion problem is a model-based extended Kalman filter [2, 9, 14, 18]. An interesting alternative to this classical approach is recursive implementations of Monte Carlo based statistical signal processing [8], which are known as particle filters, see [5]. The more non-linear model, or the more nonGaussian noise, the more potential particle filters have, especially in applications where computational power is rather cheap and the sampling rate slow. A framework for particle filtering in positioning, navigation and tracking applications is further discussed in [10]. Models of motion and measurements form a natural basis for the work and are presented in Section 2. The particle filter is introduced in Section 3, followed by simulations and a short discussion. Finally, Section 5 provides some conclusive remarks. 2
MODELS
Central for all position applications is the characterization of measurements. When considering model-based sensor fusion, also the motion model is vital. A rather generic model can be written as xt+1 = f (xt ) + Bu ut + Bw wt yt = h(xt ) + et .
(1a) (1b)
Motion models (1a) are further discussed in Section 2.1, while Section 2.2 provides measurement equations (1b).
of the signal amplitude known as fast fading, but it has little effect on the travelled distance or low-pass filtered power measurements.
2.1
The base stations in a terrestrial wireless communications system act as beacons by transmitting pilot signals of known power. The mobile station monitors the M (in GSM, M = 5, and in the upcoming WCDMA standard [1], M = 6) strongest signals, and reports regularly (or event-driven) the measurements to the network. Based on these measurements, the network centrally transfers connections from one base station to another (handover) when the mobile is moving during the service session. According to the empirical model by Okumura-Hata [11], this received power typically decays as ∼ K1 /rα , α ∈ [2, 5], where K1 and α are depending on the radio environment, antenna characteristics, terrain etc. One natural approximation is therefore to assume that K1 and α are the same for similar base stations in a service area with roughly the same terrain. All in all, in a logarithmic scale, this provides M measurement equations, one for each available base station, according to (3a) ha,i (pt ) = K − α log10 ( pi − pt + mit ),
Motion Models
The signals of primary interest in positioning applications are related to position pt , velocity vt and acceleration at . Depending on whether the signals are measureable or not, they may be components of either the state vector xt or the input signal ut . Other parameterizations, however, might provide better understanding of design variables and algorithm tuning. Depending on the context, the velocity and/or the acceleration might be measurable (the natural example is a car-mounted system) The state dynamics with/without velocity measurements can be modeled as pt+1 = pt + Ts vt + Ts wt |{z} |{z} | {z } Bu ut
xt
(2a)
Bw wt
2 I Ts · I pt pt+1 T · I s vt+1 = 0 vt + Ts · I wt I {z } | {z } | {z } | A
xt
(2b)
Bw
In (2b), the process noise wt could be multimodal, where each mode represents a specific event having a certain probability. For example, when driving in an urban environment we know that the car most of the time follows a straight line, but occasionally makes sharp turns. Of course, even more complex models are possible, e.g. the coordinated turn model [9]. 2.2
Measurement Equations
The main difference between the different positioning approaches is the measurements available. A generic model of a measurement equation is according to (1b), where the measurement noise contributions et are characterized by their distributions. If not explicitely mentioned, a Gaussian distribution is assumed. Every available measurement signal thus corresponds to a measurement equation. In this work, the focus is on measurements related to relative distance rti = |pt − pi | between the mobile position pt and a number of base station positions pi , i = 1, . . . , M . Essentially, the measurements reflect the travelled distance of radio signals, either via time measurements or power measurements. In suburban and
where K = log10 K1 . Similar measurements, but without considering or modeling the bias, are used in [13]. Point-mass implementation of estimators based on RF measurements is also discussed in [4].
To provide more accurate positioning via RF measurements, future mobile stations will be able to estimate the traveled time (and hence the distance) of radio signals from a multitude of base stations with known positions. The aforementioned methods TOA, TDOA, OTD all are based on this principle. The resulting M (M is typically 1-3) measurement equations can thus be modeled by (3b). hb,i (pt ) = |pi − pt | + mit ,
Any signaling aspects are neglected, but it is identified that network-assisted and mobile-assisted systems will result in different signaling requirements and situations. Note that any other available measurement signal relating to the position can be readily incorporated by adding a new measurement equation. Also highly nonlinear information such as map information can thus also be utilized [10]. 2.3
Figure 2: Far-field scattering and local scattering of transmitted signals in a wireless network. urban areas, the line-of-sight (LoS) path is sometimes blocked by buildings. Due to this far-field scattering, where the signals might take a non-line-of-sight path (see Figure 2), the measurements can be biased. Instead, the measurements reflect the travelled distance as rti = pi − pt + mit , i = 1, . . . , M.
The signals are also subject to near-field scattering, where the signals are reflected by a large number of objects close to the receiver. This causes fast variations
(3b)
Unknown Parameters
Positioning aims at estimating the position pt . If the velocity is unknown, it can also be incorporated in the state vector of the motion model as in (2b). Furthermore, the biases mit are unknown and can be assumed to change rather step-wise, when the signal path occasionally changes rapidly due to new reflections. A plausible state evolution model for the biases is therefore mit+1 = mit + nit , p(nit ) =
1 X
Prk N (0, σk ),
(4)
k=0
where nit takes on values from two Gaussian distributions, with a small and a large variance respectively (σ0
